1. Field of the Invention
The present invention relates to a polyphase power grid, and, more particularly, to a method for measuring stability margin at a node of a polyphase power grid.
2. Description of the Related Art
The transmission capacity of a power grid, e.g., the national power grid, is limited by factors such as voltage stability, transient stability and oscillatory stability, in addition to thermal overload conditions that result in excessive transmission line sag. The potential for oscillatory instability, also known as subsynchronous resonance, is of particular concern because the transition from stable to unstable conditions can be quite abrupt and may occur without warning. In a matter of seconds, a power system that becomes small-signal unstable can transition to a condition of large amplitude voltage, current and torque fluctuation that can be highly disruptive and destructive. Although voltage and current waveforms in the system may provide warnings of an impending instability, this is not always the case. A system that has a low margin of stability can operate very close to the threshold of instability while showing no signs of impending oscillation if there are no perturbations on the system with energy in the spectral range near the frequency of minimum stability margin. A slight change in operating conditions, such as a small increase in load, can cause the system to cross the threshold and suddenly experience violent excursions.
A primary cause of oscillatory instability is negative resistance. For example, switchmode power converters are being employed in increasing numbers and higher power capacities. The high efficiency and well regulated outputs that can be obtained with this technology bring with it an inherent negative resistance characteristic at the AC input that can cause oscillatory instability. Variable Speed Drives also offer high efficiency and well-controlled performance but with similar negative resistance characteristic. High Voltage DC (HVDC) transmission systems have desirable characteristics that avoid many of the difficulties of AC power transmission. HVDC transmission systems are members of the switchmode power converter family and have similar negative input resistance qualities. Induction motors, which may exhibit a negative input resistance, constitute the major portion of the load on the transmission system.
Methods of stability margin measurement in DC powered switchmode power converter systems are well established in the industry. A theoretical basis for measuring the margin of stability at the AC interface, the method of constructing special electronic test equipment required for such measurement and the method of performing these measurements was disclosed in “Measurement of Stability Margins in Single-Phase and Polyphase Switchmode Power Systems” (Interim Report), 31 Mar. 1997, Contractor Report to the Naval Surface Warfare Center (NSWC) Annapolis, Md., Code 814, Williams, Michael L. The method employs the use of suppressed-carrier stimulus and product demodulation of the amplitude-modulated response signals. A tutorial paper explaining the theoretical basis for his method of measuring the stability margin at AC power interfaces may be found in “Measurement of Stability Margins in Single-Phase and Polyphase Switchmode Power Systems—A Tutorial Introduction,” Proceedings, American Society of Naval Engineers Electric Machines Technology Symposium, Jan. 2004, Philadelphia, Pa., Williams, Michael L.
The suppressed-carrier waveform of FIG. 1A is formed by multiplying the 60 Hertz (Hz) power line voltage waveform of FIG. 1B by the test frequency waveform of FIG. 1C, under the assumption that the selected test frequency is 6 Hz. FIG. 1D is the amplitude modulated power line voltage or current. FIG. 1E is the demodulated voltage or current waveform formed by multiplying the waveform of FIG. 1D by the waveform of FIG. 1B.
The envelope of the suppressed-carrier waveform follows the shape of the test frequency waveform if we account for the phase reversal in the second half. The fine structure of the suppressed-carrier waveform is in-phase with the power line voltage waveform during the first half cycle of the test frequency waveform but is out-of-phase during the second half when the test frequency waveform goes negative. The positive peaks of the waveform in FIG. 1A follow the form of the 6 Hz waveform in FIG. 1C during the first half cycle, but the negative peaks follow the waveform during the second half cycle. The spectrum associated with each of the waveforms is presented. The frequency of the spectral lines of the suppressed-carrier waveform of FIG. 1A can be determined from the trigonometric identity of equation EQ. 1. When the 60 Hz power line waveform is multiplied by a 6 Hz test frequency waveform, the result is a suppressed-carrier waveform with double-sided spectral components at 60±6 Hz and −60±6 Hz.CosB CosC=½[Cos(B+C)+Cos(B−C)]  EQ. 1
In DC switchmode power converter systems, stability is determined by the complex ratio of source impedance, ZS, to the load impedance, ZL, formed by the input impedance of the converter. This method is well established in the industry. In FIG. 2 the transfer function VZL/E, shown in equation EQ. 2, becomes infinite if the complex ratio ZS/ZL becomes equal to minus one.VZL/E=1/(1+ZS/ZL)   EQ. 2VZL is an AC voltage developed across the load impedance at the test frequency, and E is the injected sinusoidal stimulus voltage. The complex ratio ZS/ZL can be plotted on a Nyquist diagram to evaluate the degree of stability. For a stable system, the closeness of approach of ZS/ZL to the point −1+j0 provides a convenient one-dimensional measure of the margin of stability. The closeness of approach is defined by the magnitude of the vector quantity 1+(ZS/ZL), which is known in circuit theory as the Return Difference. The threshold of instability occurs when 1+(ZS/ZL) becomes equal to zero.
Stability margins of DC systems are measured by injecting a sinusoidal voltage or sinusoidal current stimulus. Sinusoidal response signals are produced as the induced stimulus current interacts with the source impedance and load impedance. The information needed to determine the margin of stability is contained in the relative amplitude and phase of these two response signals. The complex ratio ZS/ZL can be determined by the ratio of voltages VZS and VZL shown in FIG. 2 (equation EQ. 3), or by the ratio of currents IL and IS (equation EQ. 4) as shown in FIG. 3 and FIG. 4.ZS/ZL=−VZS/VZL   EQ. 3ZS/ZL=IL/IS   EQ. 4
Respectively, VZS and VZL are the AC voltages developed across the source impedance and load impedance at the test frequency. Likewise, IS and IL are the AC currents flowing in the source and load impedance. The test frequency is swept over the band of interest.
FIG. 3 assumes the sinusoidal stimulus current is injected by an ideal current source. FIG. 4 assumes that the stimulus current is injected by a sinusoidal voltage source, E, having finite impedance, RA, which may be low. It is noted that the current ratio IL/IS is not altered by the impedance of the current source. The DC voltage, EA, is adjusted to minimize the DC current flowing in RA. Noted further, a system having a ZS/ZL ratio that encircles the point 1+j0 can be made stable by a sufficiently low value of the current source impedance, RA. In such cases, measurements can be made of the ratio, ZS/ZL, at points approaching the threshold of instability from both directions. Interpolation using two points on either side of the stability threshold can improve the accuracy of determining the threshold.
In DC systems, the response signals are sinusoidal and the margin of stability is determined by the relative amplitude and phase angle of the two sinusoidal responses. In AC systems, the response signals are the envelopes of the amplitude-modulated waveforms produced by the suppressed-carrier stimulus current acting on the source and load impedance. The margin of stability can be determined by the relative amplitude and phase angle of the modulation envelopes of the two response signals. In principle, the stability margin could be determined by observing the relative amplitude and phase shift of the modulation envelopes with an oscilloscope. In practice, the envelopes of the waveforms are extracted by demodulation and the relative amplitude and phase are determined with a narrowband network analyzer. This process provides a high degree of accuracy in measuring the amplitude and phase angles of the response signals. It also provides a high degree of discrimination against noise, distortion and interference. The DC case may be considered to be simply a special case of the more general AC case, a special case in which the power line frequency is zero.
The suppressed-carrier stimulus produces an amplitude modulation of the quiescent AC power waveforms. In FIG. 2, if the injected sinusoidal stimulus voltage, E, is relocated to the left side of ZS, it may be seen that the injected stimulus effectively amplitude modulates the quiescent DC voltage. With the relocation of the stimulus voltage, E, the stability margin continues to be determined by the ratio of the AC voltages VS and VL across the source and load impedance. Both the AC and DC cases involve amplitude modulation.
Consider a situation in which both the DC case and AC case are to be stimulated at the test frequency FS=6 Hz. In FIG. 1A, the spectrum of the suppressed-carrier waveform consists of modulation sidebands resulting from product modulation of the 60 Hz power line waveform of FIG. 1B by the 6 Hz test frequency waveform of FIG. 1C. The spectrum of the 6 Hz sine wave of FIG. 1C has been translated in frequency by +/60 Hz. If the power line frequency were to be reduced, the spectral pairs of FIG. 1A would remain separated by 2 FS, but they would move closer to zero. If the power line frequency is further reduced to zero, the spectral pairs of FIG. 1A converge and become identical to the spectrum of the test frequency sinusoid shown in FIG. 1C. The AC case is identical to the DC case, if the power line frequency is reduced to zero. From the perspective of amplitude modulation, the DC and AC cases are similar. From the spectral viewpoint, the AC and DC cases are similar. Conceptually, DC can be considered to be simply a single point on the continuum of positive and negative frequency.
In an unstable rotary system, experiencing mild subsynchronous oscillation at 3 Hz, the torsional shaft oscillation will occur at 3 Hz, and the system voltages and currents will be amplitude modulated with a 3 Hz envelope. There will be no 3 Hz component in the power line voltage or current waveforms. Instead, the spectral components of the oscillation will occur at 60±3 Hz.
As in the DC case, there are a number of variations in which the suppressed-carrier method of stability margin measurement may be configured. Stimulus may be provided by series voltage injection or parallel current injection. When series voltage injection of the stimulus is employed, the response signals are sensed as voltages developed across the source impedance and load impedance. When parallel current injection of the stimulus is employed, the response signals are sensed as currents flowing in the source impedance and load impedance. Series voltage injection tends to be invasive, requiring disruption of power flow to insert the injection transformers.
FIG. 5 illustrates the method of stability margin measurement in 3-phase AC systems using series voltage stimulus injection. The generator supplies power to a switchmode power converter load via source impedance, ZS. The switchmode power converter has a constant-power nature with characteristic negative resistance input. This load is represented by a Wye connected network having a per-leg impedance ZL. The blocks labeled L/N convert the three phase line voltages to line-to-neutral form in order to accommodate floating delta power. An artificial neutral point is created at the centroid of the line-to-line voltage vectors. The block labeled SCMOD contains three multipliers that form the suppressed-carrier stimulus voltages by multiplying samples of the respective line-to-neutral voltages by the test signal waveform, at frequency FS. The suppressed carrier waveforms are inserted in the associated phase via transformers driven by power amplifiers. In this application, it is assumed that the phase shift produced by the source impedance is small at the power system frequency. This allows the phase references to be taken from the undistorted generator output rather than the junction of ZS and ZL that may have severe distortion resulting from rectification harmonics.
The suppressed carrier stimulus signals cause the phase voltages, on either side of the SCMOD block, to be amplitude modulated. The response signals are in the form of these amplitude-modulated voltages on either side of the injected suppressed-carrier stimulus voltage. The blocks labeled PDMOD contain three product demodulators that multiply line-to-neutral samples of the amplitude-modulated waveforms by their respective line-to-neutral samples of the reference phase voltage. The three outputs of the PDMOD blocks have a DC component and the fine structure of the waveforms has a frequency that is two times that of the power line. These waveforms retain the modulation envelope of the amplitude-modulated waveforms. The blocks labeled + sum the outputs of the PDMOD blocks. Because the double frequency component of the PDMOD outputs exist at phase angles of 120 degrees relative to each other, they tend to sum toward zero. The output of the + blocks contains the desired sinusoid, at the test frequency, having an amplitude and phase that relates to the envelope of the amplitude-modulated waveforms. The sum also contains a DC component that is removed by capacitor coupling of the network analyzer input. The network analyzer measures the relative amplitude and phase of the two outputs of the + blocks to determine the desired value of ZS/ZL at the test frequency, FS. The test frequency is swept over the frequency range of interest and the ZS/ZL ratio is plotted as a Nyquist diagram on the network analyzer display. The margin of stability is read from the Nyquist display.
FIG. 6 illustrates a test configuration that employs stimulus in the form of suppressed-carrier current wherein the response signals are sensed as currents. In this example an auxiliary 60 Hz power source is employed to supply the bulk of the volt-amperes required for stimulus injection. The AUX power source is synchronized with the three phase voltages at the junction of ZS and ZL. When synchronized, the AUX source supplies negligible 60 Hz current to the junction of ZS and ZL. The block labeled SC MOD is the same as that in FIG. 5. It injects suppressed-carrier voltage stimulus in series with the outputs of the AUX power source. The resistors, labeled R, connect one side of the SC MOD block with the junction of ZS and ZL. In this configuration, the suppressed-carrier stimulus voltages cause suppressed-carrier stimulus currents to be injected into the junction of ZS and ZL. This set of stimulus currents is labeled I. In accordance with the current division theorem illustrated in FIG. 3 and FIG. 4, the stimulus currents divide in inverse proportion to the ratio ZS/ZL. Two sets of current transformers sense the divided stimulus currents flowing toward the prime power source and the load. After demodulation and summation, the network analyzer measures the ratio IL/IS to determine the ratio ZS/ZL in accordance with equation EQ. 4.
The test frequency, FS, is swept over the band of interest. The network analyzer measures the ratio IL/IS and plots the quantity ZS/ZL as a Nyquist diagram to determine the stability margin. The load, represented by ZL, may include a rectifier that causes significant harmonic distortion at the junction of ZS and ZL. In FIG. 6, it is assumed that the modulator/demodulator reference signals must be obtained from distorted signals at this junction. The BPF blocks provide bandpass filtering that removes harmonic distortion to provide relatively clean reference signals. Bandpass filtering produces no phase shift at the center frequency. In this situation, the predominant harmonics are likely to be 5th and 7th. These harmonics are well removed from the 60 Hz fundamental and are adequately attenuated by bandpass filters having modest Q.
The resistors in the stimulus injection path are intended to ease the problem of equalizing the AC voltage of the AUX power source with that existing at the point of stimulus injection and to minimize the flow of harmonic distortion currents in the injection path. In the choice of the value of the resistors, R, it should be remembered that the stability margin measurement is a small-signal measurement. If there is significant harmonic distortion at the junction point resulting from rectification harmonics, the resistance values should be sufficiently large to avoid altering the commutation timing of the rectification process. Larger values of resistance require a larger volt-ampere output of the power amplifiers used for stimulus injection. At lower power levels, amplifiers are available that have the capability of providing sufficient power for stimulus injection.
The data points obtained from the stability margin measurement are combined and presented as a Nyquist diagram. The stability margin is determined by the closeness of approach of the Nyquist plot to the point 1+j0.
Substantial obstacles exist in the practical application of the above-described methods in the measurement of stability margin and/or impedance at nodes of the national power grid. These methods were developed to measure the stability margin at the AC power interface of switched-mode power converter systems operating at power levels of less than approximately 100 KW, whereas the national power grid, or portions thereof, operate at hundreds or thousands of megawatts.
For example, switchmode power converters normally have significant bandwidth in their regulator control loops that causes the frequency of potential instability to occur at frequencies that are well separated from the power line frequency, whereas oscillatory instability of the national power grid generally occurs at frequencies that lie in the range of 0.1 Hz to 3.0 Hz, thereby producing modulation sidebands that must be measured at frequencies that are separated from the power line frequency by an amount of only 0.17 percent to 5.0 percent. The close proximity to the power line frequency requires their measurement to be made in the frequency region of significant phase noise existing near the shifting power line frequency.
Also, for example, for switchmode power converter systems, stimulus injection is normally provided by power amplifiers having sufficient capacity to supply suppressed-carrier stimulus injection on the order of 0.03 Per Unit, which is an order of magnitude higher than the level of stimulus injection that would be permissible for application in determining stability margin and/or impedance at nodes of the national power grid.